Showing papers on "Constant (mathematics) published in 1984"
TL;DR: In this paper, a method is described to realize coupling to an external bath with constant temperature or pressure with adjustable time constants for the coupling, which can be easily extendable to other variables and to gradients, and can be applied also to polyatomic molecules involving internal constraints.
Abstract: In molecular dynamics (MD) simulations the need often arises to maintain such parameters as temperature or pressure rather than energy and volume, or to impose gradients for studying transport properties in nonequilibrium MD A method is described to realize coupling to an external bath with constant temperature or pressure with adjustable time constants for the coupling The method is easily extendable to other variables and to gradients, and can be applied also to polyatomic molecules involving internal constraints The influence of coupling time constants on dynamical variables is evaluated A leap‐frog algorithm is presented for the general case involving constraints with coupling to both a constant temperature and a constant pressure bath
25,256 citations
TL;DR: In this article, the authors studied square integrable coefficients of an irreducible representation of the non-unimodular $ax + b$-group and obtained explicit expressions in the case of a particular analyzing family that plays a role analogous to coherent states (Gabor wavelets) in the usual $L_2 $ -theory.
Abstract: An arbitrary square integrable real-valued function (or, equivalently, the associated Hardy function) can be conveniently analyzed into a suitable family of square integrable wavelets of constant shape, (i.e. obtained by shifts and dilations from any one of them.) The resulting integral transform is isometric and self-reciprocal if the wavelets satisfy an “admissibility condition” given here. Explicit expressions are obtained in the case of a particular analyzing family that plays a role analogous to that of coherent states (Gabor wavelets) in the usual $L_2 $ -theory. They are written in terms of a modified $\Gamma $-function that is introduced and studied. From the point of view of group theory, this paper is concerned with square integrable coefficients of an irreducible representation of the nonunimodular $ax + b$-group.
3,423 citations
TL;DR: In this article, a first-order perturbation approximation of the Eulerian velocity covariances for uniform average flow is used to derive closed-form expressions of the closedform expressions for the concentration expectation value, which satisfies a diffusion equation with time-dependent apparent dispersion coefficients.
Abstract: Solute transport in porous formations is governed by the large-scale heterogeneity of hydraulic conductivity. The two typical lengthscales are the local one (of the order of metres) and the regional one (of the order of kilometres). The formation is modelled as a random fixed structure, to reflect the uncertainty of the space distribution of conductivity, which has a lognormal probability distribution function. A first-order perturbation approximation, valid for small log-conductivity variance, is used in order to derive closed-form expressions of the Eulerian velocity covariances for uniform average flow. The concentration expectation value is determined by using a similar approximation, and it satisfies a diffusion equation with time-dependent apparent dispersion coefficients. The longitudinal coefficients tend to constant values in both two- and three-dimensional flows only after the solute body has travelled a few tens of conductivity integral scales. This may be an exceedingly large distance in many applications for which the transient stage prevails. Comparison of theoretical results with recent field experimental data is quite satisfactory.The variance of the space-averaged concentration over a volume V may be quite large unless the lengthscale of the initial solute body or of V is large compared with the conductivity integral scale. This condition is bound to be obeyed for transport at the local scale, in which case the concentration may be assumed to satisfy the ergodic hypothesis. This is not generally the case at the regional scale, and the solute concentration is subjected to large uncertainty. The usefulness of the prediction of the concentration expectation value is then quite limited and the dispersion coefficients become meaningless.In the second part of the study, the influence of knowledge of the conductivity and head at a set of points upon transport is examined. The statistical moments of the velocity and concentration fields are computed for a subensemble of formations and for conditional probability distribution functions of conductivity and head, with measured values kept fixed at the set of measurement points. For conditional statistics the velocity is not stationary, and its mean and variance vary throughout the space, even if its unconditional mean and variance are constant. The main aim of the analysis is to examine the reduction of concentration coefficient of variation, i.e. of its uncertainty, by conditioning. It is shown that measurements of transmissivity on a grid of points can be effective in reducing concentration variance, provided that the distance between the points is smaller than two conductivity integral scales. Head conditioning has a lesser effect upon variance reduction.
925 citations
TL;DR: In this paper, a nonrelativistic potential theory for gravity is proposed, which is built on the basic assumptions of the modified dynamics, which were shown earlier to reproduce dynamical properties of galaxies and galaxy aggregates without having to assume the existence of hidden mass.
Abstract: We consider a nonrelativistic potential theory for gravity which differs from the Newtonian theory. The theory is built on the basic assumptions of the modified dynamics, which were shown earlier to reproduce dynamical properties of galaxies and galaxy aggregates without having to assume the existence of hidden mass. The theory involves a modification of the Poisson equation and can be derived from a Lagrangian. The total momentum, angular momentum, and (properly defined) energy of an isolated system are conserved. The center-of-mass acceleration of an arbitrary bound system in a constant external gravitational field is independent of any property of the system. In other words, all isolated objects fall in exactly the same way in a constant external gravitational field (the weak equivalence principle is satisfied). However, the internal dynamics of a system in a constant external field is different from that of the same system in the absence of the external field, in violation of the strong principle of equivalence. These two results are consistent with the phenomenological requirements of the modified dynamics. We sketch a toy relativistic theory which has a nonrelativistic limit satisfying the requirements of the modified dynamics.
887 citations
TL;DR: In this paper, the LJ fluid at a density and temperature close to the triple point is determined and compared for molecular dynamics computer simulations using (N, V, E), (N V, T, T), ( N, P, H) and (N P, T) ensembles.
Abstract: Dynamic and static properties of the LJ fluid at a density and temperature close to the triple point are determined and compared for molecular dynamics computer simulations using (N, V, E), (N, V, T), (N, P, H) and (N, P, T) ensembles. As expected, the mean values of thermodynamic properties for all the different ensembles show good agreement. For the velocity autocorrelation function and the time dependence of the mean square displacement it is shown that the constant temperature and/or constant pressure algorithms used produce results which are identical, within statistical accuracy, to those obtained using the constant energy ensemble. The equations of motion are presented in a readily implementable form.
325 citations
TL;DR: In this article, the authors studied functional and functional differential equations and showed that functional equations are directly connected with difference equations of a discrete (for example, integer-valued) argument, the theory of which has been very intensively developed in the book and in numerous subsequent papers.
315 citations
TL;DR: In this article, Goldberger and Kmenta used a logarithmic demand curve fitted to monthly data to measure the influence of categorical variables in regression equations, and showed that the results presented in the usual way involve no special problems of interpretation, and that the difference between the purely mechanical problem of fitting the regression and the quite different problem of presenting the results in the most effective fashion is important.
Abstract: Regressions containing dummy variables are easily estimated by the familiar expedient of "dropping out" one of the categories but the result is often awkward to interpret. Since coefficients of dummy variables are determined only up to an additive constant, however, the equation can be transformed into a more easily interpretable form by adding on an appropriately chosen constant to each coefficient. For most regressions the constants should be chosen to force the mean of the transformed coefficients to equal 0. For logarithmic regressions the constants should be chosen to force the sum of the antilogs of the coefficients to equal 1. With logarithmic demand curves fitted to monthly data the resulting antilogs become monthly seasonal indexes. The technical procedure by which dummy variables are used to capture the influence of categorical variables in regression equations is generally familiar (see Goldberger (1964), Kmenta (1971), Johnston (1960), or, to go back near the beginning of things, Suits (1957)). In many cases, particularly where only two classes of observation are involved, results presented in the usual way involve no special problems of interpretation. For example, use of a dummy variable to distinguish pre-war from post-war behavior, or to measure the shift in a relationship during the period of a strike is readily understood by any reader. But where a set of several dummy variables is employed to measure the variation in behavior among a number of classes-regions, education groups, age brackets, and the like-there is often an important difference between the purely mechanical problem of fitting the regression and the quite different problem of presenting the results in the most effective fashion. The purpose of this paper is to call attention to this distinction, and to illustrate by simple examples.
278 citations
01 Jan 1984
TL;DR: In this paper, a maximin formula for the directional derivative of the marginal value function of a perturbed nonlinear mathematical programming problem is obtained under the constant rank regularity assumption.
Abstract: Under the constant rank regularity assumption, a maximin formula is obtained for the directional derivative of the marginal value function of a perturbed nonlinear mathematical programming problem.
197 citations
TL;DR: In this article, the intersection of the range of the operator d2/dt2 + c(d/dt) f u sin(.) acting on Znperiodic functions of class g* with the subspace of constant functions in the space C([O, 2711) of real continuous functions on [0,27r] is the closed interval [--a, a], whose interior points are images of two distinct solutions and boundary points of one.
163 citations
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TL;DR: In this article, the authors present an approximate linearized framework for analysis of the term structure in which these differences disappear, and test its accuracy in practice using data from the CRSP government bond tapes.
Abstract: Expectations theories of asset returns may be interpreted as stating either that risk premia are zero, or that they are constant through time. Under the former interpretation, different versions of the expectations theory of the term structure are inconsistent with one another, but I show that this does not necessarily carry over to the constant risk premium interpretation of the theory. Furthermore, I argue that differences among expectations theories are of 'second order" in a precise mathematical sense. I present an approximate linearized framework for analysis of the term structure in which these differences disappear, and I test its accuracy in practice using data from the CRSP government bond tapes.
148 citations
01 Dec 1984
TL;DR: Stockmeyer [St] showed that probabilistic bounded depth circuits can approximate the exact number of ones in the input with very low probability of error.
Abstract: 1. In t roduc t ion I{t~txml, ly there has been much interest ill tht, comput:Ltional power of circuits o1' bounded dt;pth. In particular Furst, Saxe and Sipser [FSS], and indcpentlently Ajtai [Aj] in a different form, have shown that no polynomial size circuits of bounded depth can compute the parity function of n boolean variables (and other related functions such as the majority function, the exact number of ones in the input, etc.). On the other hand, Stockmeyer [St] showed that probabilistic bounded depth circuits can approximate the exact number of ones in the input with very low probability of error. Tlmse results lead to the following interesting question: (*) Are probabilistic constant depth circuits more powerful than deterministic ones? While it is well known [BG] that (non uniform) deterministic polynomial size circuits are as powerful as probabilistic ones, the question is still open for bounded depth circuits. This is so because the reduction given by Bennet and Gill [BG]
TL;DR: In this paper, the authors derived the E.O. for constant, continuous demand for linear increasing demand, and extended the time horizon so that it no longer influences the replenishment times.
Abstract: The E.O.Q. is derived for constant, continuous demand. For linear increasing demand, Donaldson's analytical solution considers demand bounded by a time horizon H. Although this is mathematically convenient for obtaining a solution, it complicates the calculation of the optimal replenishment policy. Extending the time horizon so that it no longer influences the replenishment times simplifies the calculation of the optimal policy, which is then equivalent to the E.O.Q. calculation for constant demand.
TL;DR: In this article, the diffusion equation for a constant and a linear potential is solved with boundary conditions which account for back reaction (desorption), and the solution is given in terms of Green's function, from which expressions for the survival probability are derived.
Abstract: The diffusion equation for a constant and a linear potential is solved with boundary conditions which account for back‐reaction (desorption). The solution is given in terms of Green’s function, from which expressions for the survival probability are derived. Inclusion of back reaction generally results in an ultimate survival probability of unity.
01 Jan 1984
TL;DR: In this article, the authors discuss the issues that arise in the design of blending facilities for solid modellers, summarizes proposed approaches, and presents a new approach for incorporating constant-size radius blends, conceptually generated by rolling a sphere in contact with the surfaces to be blended, in modellerers based on constructive solid geometry (CSG).
Abstract: Current solid modellers lack facilities for describing blends, i.e., smooth fillets and rounds between surface features of mechanical parts. This paper discusses the issues that arise in the design of blending facilities for solid modellers, summarizes proposed approaches, and presents a new approach for incorporating constant radius blends, conceptually generated by rolling a sphere in contact with the surfaces 1i9 be blended, in modellers based on constructive solid geometry (CSG). Blending is a catch-all term used to denote the creation of fillets, rounds, and similar smooth, localized transitions between large-scale surface features of a solid object. The major types of blends encountered in practice can be categorized as follows: • Surfaces governed by strong Junctiona.l constraints. For example, the surface joining the wing of an aircraft to the fuselage must meet stringent aerodynamic requirements. • Esthetic blends. For example, the smooth transition surface between the body and the bottom of a wine glass is constrained by esthetic considerations, rather than by mechanical function. • Fairings [30], i.e., transition surfaces that are relatively large (when compared to the surface features being blended), and whose shape is neither strongly constrained by function nor by esthetics. Typically, fairings serve to connect functional features such as bearing housings. Many examples are found in automobile suspension parts, ducts and manifolds. • Rounds and fillets. These are relatively small transition surfaces whose shape is weakly constrained by function. They are found in almost all machined, cast or molded parts, and may serve to relieve stress
TL;DR: It is proved to never expand more nodes than B or A∗ and to expand a much smaller number of them in some cases and a proof that no overall optimal algorithm exists if the cost of an algorithm is measured by the total number of node expansions.
TL;DR: In this paper, the equations of an electrically conducting compressible fluid in electro-magneto-fluid dynamics are studied and it is proved that in a certain case of two-dimensional flow, the fluid become a symmetric hyperbolic-parabolic system in both of the viscous and non-viscous cases.
Abstract: The equations of an electrically conducting compressible fluid in electro-magneto-fluid dynamics are studied. It is proved that in a certain case of two-dimensional flow, the equations of the fluid become a symmetric hyperbolic-parabolic system in both of the viscous and non-viscous cases. Therefore, the initial value problem is well posed in the Sobolev spaces at least for short time interval. Furthermore, in the viscous case, the solution exists globally in time and tends to the constant state as time goes to infinity, provided the initial data are closed to the constant state. The proof is based on a technical energy method, which makes use of a quadratic function associated with the total energy of the fluid.
TL;DR: In this article, conditions which may lead to freezing of the motion of a system under continuous observation (the so-called "Zeno paradox" or "watchdog effect") are examined.
Abstract: Conditions which may lead to a freezing of the motion of a system under continuous observation (the so-called "Zeno paradox" or "watchdog effect") are examined. The measurement process is treated phenomenologically by the usual wave-packet reduction as well as in a more realistic way by including the measuring apparatus. For this purpose a model for an ideal measurement process is employed, following an example given by von Neumann. The resulting behavior varies between complete freezing and a mere suppression of interference terms and constant transition rates as represented by a master equation (rate equation). The most familiar example of the latter is Fermi's golden rule, with integration leading to exponential decay. Reviewing and extending the derivation of the Pauli master equation, the conditions leading to constant transition rates are discussed. The importance of the interaction with the natural environment for establishing a master equation is emphasized. Some consequences for the derivation of macroscopic equations of motion and for the physical foundations of superselection rules are pointed out.
TL;DR: A practical approach to the problem of simultaneously computing a function, its partial derivatives with respect to all the variables, and an estimate of the rounding error incurred in the computed value of the function, in a form easily implementable as a computer program.
Abstract: A practical approach is proposed to the problem of simultaneously computing a function, its partial derivatives with respect to all the variables, and an estimate of the rounding error incurred in the computed value of the function. Theoretically, it has a complexity at most a constant times as large as that of evaluating the function alone, the constant being independent of the number of variables of the function, and it is an alternative graphical interpretation of W. Baur and V. Strassen’s results, with some generalizations. Practically, it is stated in a form easily implementable as a computer program, which enables us to automatically compute the derivatives if we are given only the program for computing the function. Remarks are added also on the cases of several functions, of higher derivatives and of non-straght-line programs, and on application to problems containing differential equations.
TL;DR: It is known that the Riemann hypothesis implies for all k ≥ 0 (see Ramachandra [2] and Heath-Brown [1]), but there is not even a conjectural value of ck for any other k as mentioned in this paper.
Abstract: LetIt is conjectured that for any k,for some constant ck. It is well known that (2) holds for k = 0, 1 and 2 with c0 = 1, C1 = 1, and c2 = (2π2)-1, but there is not even a conjectural value of ck for any other k. However, it is known that the Riemann hypothesis impliesfor all k ≥ 0 (see Ramachandra [2] and Heath-Brown [1]).
TL;DR: In this paper, a prototype of Rastall's theory of gravity, in which the divergence of the energy-momentum tensor is proportional to the gradient of the scalar curvature, is derived from a variational principle.
Abstract: A prototype of Rastall’s theory of gravity, in which the divergence of the energy-momentum tensor is proportional to the gradient of the scalar curvature, is shown to be derivable from a variational principle. Both the proportionality factor and the unrenormalized gravitational constant are found to be covariantly constant, but not necessarily constant. The prototype theory is, therefore, a gravitational theory with variable gravitational constant.
TL;DR: On calcule la mesure pour laquelle une suite de polynomes avec une formule de recurrence constante est orthogonale as discussed by the authors, et al.
TL;DR: An algorithm for determining whether two polyhedra are congruent is described and it is shown that under some conditions the problem of partial congruity can be solved in O ( n 2 ) time.
TL;DR: In this paper, the bilocal approximation in strong fluctuation theory leads to an expression for the effective dielectric constant of a random medium, and a detailed study of this expression is made for the case of a medium characterized by an isotropic correlation function.
Abstract: The bilocal approximation in strong fluctuation theory leads to an expression for the effective dielectric constant of a random medium. A detailed study of this expression is made for the case of a medium characterized by an isotropic correlation function. The effective dielectric constant is shown to obey an equation involving an integral containing the correlation function, the free space propagation constant, the quasistatic dielectric constant, and the effective dielectric constant itself, Numerical methods for solution are suggested. The low frequency behavior is also examined. The leading correction term to the quasistatic dielectric constant is shown to be different than previously deduced expressions.
TL;DR: A generalization of the EOQ formula with backorders is derived and ranges for the decision variables are obtained in this article, where the results are illustrated with the case of uniformly distributed lead time.
Abstract: This article considers an inventory model with constant demand and stochastic lead times distributed over a finite range. A generalization of the EOQ formula with backorders is derived and ranges for the decision variables are obtained. The results are illustrated with the case of uniformly distributed lead time.
TL;DR: A new method is presented for estimating the number of tuples satisfying a condition of the type attribute rel constant, where rel is one of "=", ">", "<, ", "".
Abstract: We present a new method for estimating the number of tuples satisfying a condition of the type attribute rel constant, where rel is one of "=", ">", "<", "", "". Our method gives highly accurate, y...
TL;DR: In this paper, the authors investigated some differential geometry methods in the theory of the nonlinear wave equation ∇2u=Φ(u,(∇u ∇u) ) and showed that levels of such solutions form in the space of independent variable's hypersurfaces with all principal curvatures constant.
Abstract: In this paper are investigated some differential geometry methods in the theory of the nonlinear wave equation ∇2u=Φ(u,(∇u‖∇u)). A special class of solutions is discussed for which (∇u‖∇u) is constant on each level of the function u. It is proved that levels of such solutions form in the space of independent variable’s hypersurfaces with all principal curvatures constant. The general form of such hypersurfaces is given. Then it is proved that via the method of characteristics it is possible to construct (in principle) all the solutions of the discussed class. They may be obtained by integration of an ODE of second order using a special class of the polynomial functions. Some new solutions are given for equations⧠v=4Av3+3Bv2+2cv+D, ⧠v=μ exp v, ⧠v=sin v, ⧠v=cosh v, and ⧠v=sinh v.
TL;DR: In this paper, a wave with constant form and constant velocity c on the surface of an incompressible, inviscid fluid over a horizontal bottom is assumed to be irrotational and if h is the depth of the fluid at infinity and g the acceleration due to gravity, then the Froude number F is defined by F squared = (c squared)/gh.
Abstract: : This paper is concerned with the problem of a solitary wave moving with constant form and constant velocity c on the surface of an incompressible, inviscid fluid over a horizontal bottom. The motion is assumed to be two-dimensional and irrotational, and if h is the depth of the fluid at infinity and g the acceleration due to gravity, then the Froude number F is defined by F squared = (c squared)/gh. The result that F 1 has recently been proved by Amick and Toland by means of a long and complicated argument. Here we give a short and simple one. (Author)
TL;DR: The best lower bound known for f(n) is due to Moser as discussed by the authors, who showed that f n > cn 5 7 for some fixed constant c. This lower bound was later improved to f n 2 3 (2) 9 3.
TL;DR: An analysis of the Babuska stability of bilinear/constant finite element pairs for viscous flow calculations is given in this paper, where an unstable mode not of the checkerboard type is given for which the stability constant turns out to be O(h).
Abstract: An analysis of the Babuska stability of bilinear/constant finite element pairs for viscous flow calculations is given. An unstable mode not of the checkerboard type is given for which the stability constant turns out to beO(h). Thus, the indicated spaces are not stable in general for numerical calculation.